On Thursday, December 1, 2016 at 8:14:04 AM UTC-5, radam...@gmail.com wrote:
> I wrote a paper in '92 about the repeated missing sample case. 
> 
> "Nonuniform Sampling of Audio Signals"
> 

and it had a very satisfying result relating the usable bandwidth and the average sample rate.

i remember this from nearly a quarter century ago.

r b-j

I wrote a paper in 92 about the repeated missing sample case. 

"Nonuniform Sampling of Audio Signals"

Available in the AES E-library 


Bob 

Am 30.11.16 um 10:19 schrieb kurtulmehtap@gmail.com:
> I wonder if I can sample a signal using an ADC with variable sampling frequency.
>
> Lets say the signal bandwidth is 10 MHz at baseband. In order to reconstruct the signal, the ADC sampling frequency must be >20 MHz.
>
> What if I use a sampling frequency that varies between 19.9 MHz and 20.1 MHz?
> What does the Nyquist sampling theorem tells us about this case?
>
> Are there applications that actually use variable frequency sampling?
>
> Forgive my curiosity, Thanks.
>

There is a theory for sampling a signal with non-regularl spaced 
sampling points. A classic algorithm for reconstruction of a signal is 
the Voronoi-Allebach algorithm, where you repeatedly set the missing 
samples to 0 and cut off the high frequencies in the frequency domain. 
An introduction to the topic can be found here:

https://www.math.ucdavis.edu/~strohmer/research/sampling/irsampl.html

This guy has done some quite impressive stuff with it, for example 
restoring clipped audio signals.

https://www.math.ucdavis.edu/~strohmer/research/audio/audio.html

	Christian

On Wed, 30 Nov 2016 01:19:18 -0800, kurtulmehtap wrote:

> I wonder if I can sample a signal using an ADC with variable sampling
> frequency.
> 
> Lets say the signal bandwidth is 10 MHz at baseband. In order to
> reconstruct the signal, the ADC sampling frequency must be >20 MHz.
> 
> What if I use a sampling frequency that varies between 19.9 MHz and 20.1
> MHz?
> What does the Nyquist sampling theorem tells us about this case?

The Nyquist sampling theorem tells us that for a signal bandwidth of B 
Hz, you need more than 2 * B unique samples per second to get enough 
information to accurately reconstruct the signal.

In its most absurd interpretation, this means that you could take 
20000001 samples at 1GHz (or 10, or 100) each second, and have enough 
information to reconstruct a strictly bandlimited signal.  Minor 
technical problems like the existence of noise and clock jitter, and the 
absence of perfect brick-wall filters, make this impossible -- but in 
theory, it could be done.

In practice, the "bandwidth" of the Nyquist-Shannon sampling theorem 
means something different from the "bandwidth" of a filter that you 
design, or buy off the shelf.  The Nyquist bandwidth of a signal is the 
band in which the signal energy is non-zero; alas, any such signal has a 
time response that extends to infinity, which can make waiting for any 
real-world results kinda boring.

My intuitive feel for how this would play out is thus:

1: It would be hard to get it to not alias.  Not necessarily impossible, 
but hard.

2: The long-term average sampling rate would need to exceed 20MHz.

3: "Long term" would have to be some healthy fraction of the signal's 
settling time -- if the sampling variation takes longer than it takes for 
the signal's components to settle, then you'll alias during that time 
that the sampling rate is low.

4: The math needed to figure out how to accurately reconstruct the signal 
would be torturous.

> Are there applications that actually use variable frequency sampling?

Anything that samples in the real world uses variable frequency 
sampling.  Search on "clock jitter".  I know of no applications that 
intentionally use variable frequency sampling on the input side.  On the 
output side, varying the frequency of the PWM generator in a switched-
power application is a fairly common dodge for EMI reduction in 
environments where you want to avoid having a bunch of energy 
concentrated on one "spike" in the spectrum.  Efforts to make this happen 
can range from just building a crappy oscillator to intentionally 
building in some sort of spread-spectrum modulator.

-- 
Tim Wescott
Control systems, embedded software and circuit design
I'm looking for work!  See my website if you're interested
http://www.wescottdesign.com

On Wednesday, November 30, 2016 at 10:19:23 PM UTC+13, kurtul...@gmail.com wrote:
> I wonder if I can sample a signal using an ADC with variable sampling frequency.
> 
> Lets say the signal bandwidth is 10 MHz at baseband. In order to reconstruct the signal, the ADC sampling frequency must be >20 MHz.
> 
> What if I use a sampling frequency that varies between 19.9 MHz and 20.1 MHz?
> What does the Nyquist sampling theorem tells us about this case?
> 
> Are there applications that actually use variable frequency sampling?
> 
> Forgive my curiosity, Thanks.

Provided you don't alias the signal you could vary the sampling rate but it wouldn't be such a good idea. There is a thing called random sampling where you don't need an anti-aliasing filter. I don't think it has ever been used!

On Wednesday, November 30, 2016 at 7:31:49 AM UTC-5, Steve Pope wrote:
> <kurtulmehtap@gmail.com> wrote:
> 
> >I wonder if I can sample a signal using an ADC with variable sampling frequency.
> >
> >Lets say the signal bandwidth is 10 MHz at baseband. In order to
> >reconstruct the signal, the ADC sampling frequency must be >20 MHz.
> >
> >What if I use a sampling frequency that varies between 19.9 MHz and 20.1 MHz?
> >What does the Nyquist sampling theorem tells us about this case?
> 
> It is equivalent to uniformly sampling a signal to which frequency
> modulation has been applied.  Since the latter process creates
> sidelobes stretching to infinity, some of these sidelobes will spill over 
> the original 10 KHz bandwidth, and you are no longer satisfying Nyquist.
> 
> Whether this causes actual degradation or is ignorable depends upon
> scenario.
> 
> Steve

FM on the sample clock is not exactly the same as FM on the sampled signal.


FM on the sampled clock can intuitively be thought of as speeding up and slowing down a tape playing the signal. It is more like Doppler which stretches and compresses a signal on the frequency axis, not shifts it.  Doppler SHIFT is a misnomer.

 All frequency components of the signal will move PROPORTIONALLY to their frequency but not absolutely the same amount of Hz.  

If the sampling clock changes from say 10 MHz to 10.1 MHz a component of the sampled signal at 1 MHz will move up to 1.01Mz and a component at 100 kHz will move up to 101 kHz.  Clearly the __deviation__ is not the same in absolute terms. 
So it is not the same as actual FM.  

Of course if the sampling clock at the D/a and A/D are BOTH shifted together, then nothing happens.  (ignoring differences in the delay paths)


m

 

<kurtulmehtap@gmail.com> wrote:

>I wonder if I can sample a signal using an ADC with variable sampling frequency.
>
>Lets say the signal bandwidth is 10 MHz at baseband. In order to
>reconstruct the signal, the ADC sampling frequency must be >20 MHz.
>
>What if I use a sampling frequency that varies between 19.9 MHz and 20.1 MHz?
>What does the Nyquist sampling theorem tells us about this case?

It is equivalent to uniformly sampling a signal to which frequency
modulation has been applied.  Since the latter process creates
sidelobes stretching to infinity, some of these sidelobes will spill over 
the original 10 KHz bandwidth, and you are no longer satisfying Nyquist.

Whether this causes actual degradation or is ignorable depends upon
scenario.

Steve

I wonder if I can sample a signal using an ADC with variable sampling frequency.

Lets say the signal bandwidth is 10 MHz at baseband. In order to reconstruct the signal, the ADC sampling frequency must be >20 MHz.

What if I use a sampling frequency that varies between 19.9 MHz and 20.1 MHz?
What does the Nyquist sampling theorem tells us about this case?

Are there applications that actually use variable frequency sampling?

Forgive my curiosity, Thanks.